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In mathematics, the **Virasoro algebra** is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory.

The **Virasoro algebra** is spanned by **generators** *L _{n}* for

The factor of is merely a matter of convention. For a derivation of the algebra as the unique central extension of the Witt algebra, see derivation of the Virasoro algebra.

The Virasoro algebra has a presentation in terms of two generators (e.g. L_{3} and L_{−2}) and six relations.^{[1]}^{[2]}

The generators are called annihilation modes, while are creation modes. A basis of creation generators of the Virasoro algebra's universal enveloping algebra is the set

For , let , then .

In any indecomposable representation of the Virasoro algebra, the central generator of the algebra takes a constant value, also denoted and called the representation's central charge.

A vector in a representation of the Virasoro algebra has **conformal dimension** (or conformal weight) if it is an eigenvector of with eigenvalue :

An -eigenvector is called a **primary state** (of dimension ) if it is annihilated by the annihilation modes,

A highest weight representation of the Virasoro algebra is a representation generated by a primary state .
A highest weight representation is spanned by the -eigenstates . The conformal dimension of is , where is called the **level** of .
Any state whose level is not zero is called a **descendant state** of .

For any , the Verma module of central charge and conformal dimension is the representation whose basis is , for a primary state of dimension .
The Verma module is
the largest possible highest weight representation.
The Verma module is indecomposable, and for generic values of it is also irreducible. When it is reducible, there exist other highest weight representations with these values of , called **degenerate representations**, which are quotients of the Verma module. In particular, the unique irreducible highest weight representation with these values of is the quotient of the Verma module by its maximal submodule.

A Verma module is irreducible if and only if it has no singular vectors.

A singular vector or null vector of a highest weight representation is a state that is both descendant and primary.

A sufficient condition for the Verma module to have a singular vector is for some , where

Then the singular vector has level and conformal dimension

Here are the values of for , together with the corresponding singular vectors, written as for the primary state of :

Singular vectors for arbitrary may be computed using various algorithms.^{[3]}^{[4]}

If , then has a singular vector at level if and only if with . If , there can also exist a singular vector at level if with and . This singular vector is now a descendant of another singular vector at level .

The integers that appear in are called **Kac indices**. It can be useful to use non-integer Kac indices for parametrizing the conformal dimensions of Verma modules that do not have singular vectors, for example in the critical random cluster model.

For any , the involution defines an automorphism of the Virasoro algebra and of its universal enveloping algebra.
Then the **Shapovalov form** is the symmetric bilinear form on the Verma module such that , where the numbers are defined by
and .
The inverse Shapovalov form is relevant to computing Virasoro conformal blocks, and can be determined in terms of singular vectors.^{[5]}

The determinant of the Shapovalov form at a given level is given by the **Kac determinant formula**,^{[6]}

where is the partition function, and is a positive constant that does not depend on or .

If , a highest weight representation with conformal dimension has a unique Hermitian form such that the Hermitian adjoint of is and the norm of the primary state is one. In the basis , the Hermitian form on the Verma module has the same matrix as the Shapovalov form , now interpreted as a Gram matrix.

The representation is called **unitary** if that Hermitian form is positive definite.
Since any singular vector has zero norm, all unitary highest weight representations are irreducible.
An irreducible highest weight representation is unitary if and only if

- either with ,
- or with

Daniel Friedan, Zongan Qiu, and Stephen Shenker showed that these conditions are necessary,^{[7]} and Peter Goddard, Adrian Kent, and David Olive used the coset construction or GKO construction (identifying unitary representations of the Virasoro algebra within tensor products of unitary representations of affine Kac–Moody algebras) to show that they are sufficient.^{[8]}

The character of a representation of the Virasoro algebra is the function

The character of the Verma module is

where is the Dedekind eta function.

For any and for , the Verma module is reducible due to the existence of a singular vector at level . This singular vector generates a submodule, which is isomorphic to the Verma module . The quotient of by this submodule is irreducible if does not have other singular vectors, and its character is

Let with and coprime, and and . (Then is in the Kac table of the corresponding minimal model). The Verma module has infinitely many singular vectors, and is therefore reducible with infinitely many submodules. This Verma module has an irreducible quotient by its largest nontrivial submodule. (The spectrums of minimal models are built from such irreducible representations.) The character of the irreducible quotient is

This expression is an infinite sum because the submodules and have a nontrivial intersection, which is itself a complicated submodule.

In two dimensions, the algebra of local conformal transformations is made of two copies of the Witt algebra. It follows that the symmetry algebra of two-dimensional conformal field theory is the Virasoro algebra. Technically, the conformal bootstrap approach to two-dimensional CFT relies on Virasoro conformal blocks, special functions that include and generalize the characters of representations of the Virasoro algebra.

Since the Virasoro algebra comprises the generators of the conformal group of the worldsheet, the stress tensor in string theory obeys the commutation relations of (two copies of) the Virasoro algebra. This is because the conformal group decomposes into separate diffeomorphisms of the forward and back lightcones. Diffeomorphism invariance of the worldsheet implies additionally that the stress tensor vanishes. This is known as the Virasoro constraint, and in the quantum theory, cannot be applied to all the states in the theory, but rather only on the physical states (compare Gupta–Bleuler formalism).

There are two supersymmetric *N* = 1 extensions of the Virasoro algebra, called the Neveu–Schwarz algebra and the Ramond algebra. Their theory is similar to that of the Virasoro algebra, now involving Grassmann numbers. There are further extensions of these algebras with more supersymmetry, such as the *N* = 2 superconformal algebra.

W-algebras are associative algebras which contain the Virasoro algebra, and which play an important role in two-dimensional conformal field theory. Among W-algebras, the Virasoro algebra has the particularity of being a Lie algebra.

The Virasoro algebra is a subalgebra of the universal enveloping algebra of any affine Lie algebra, as shown by the Sugawara construction. In this sense, affine Lie algebras are extensions of the Virasoro algebra.

The Virasoro algebra is a central extension of the Lie algebra of meromorphic vector fields with two poles on a genus 0 Riemann surface.
On a higher-genus compact Riemann surface, the Lie algebra of meromorphic vector fields with two poles also has a central extension, which is a generalization of the Virasoro algebra.^{[9]} This can be further generalized to supermanifolds.^{[10]}

The Virasoro algebra also has vertex algebraic and conformal algebraic counterparts, which basically come from arranging all the basis elements into generating series and working with single objects.

The Witt algebra (the Virasoro algebra without the central extension) was discovered by É. Cartan^{[11]} (1909). Its analogues over finite fields were studied by E. Witt in about the 1930s.

The central extension of the Witt algebra that gives the Virasoro algebra was first found (in characteristic *p* > 0) by R. E. Block^{[12]} (1966, page 381) and independently rediscovered (in characteristic 0) by I. M. Gelfand and Dmitry Fuchs^{[13]} (1969).

The physicist Miguel Ángel Virasoro^{[14]}
(1970) wrote down some operators generating the Virasoro algebra (later known as the **Virasoro operators**) while studying dual resonance models, though he did not find the central extension. The central extension giving the Virasoro algebra was rediscovered in physics shortly after by J. H. Weis, according to Brower and Thorn^{[15]} (1971, footnote on page 167).

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